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NUMBERS(
A number is a mathematical object used to count and measure. A notational
symbol that represents a number is called a numeral but in common use, the word
number can mean the abstract object, the symbol, or the word for the number.
Number is first discovered in India and ten symbols are used, namely o, 1, 2, 3, 4,
5, 6, 7, 8 and 9 to represent any number. These symbols are called digits.
A group of digits, denoting a number, is called a numeral. Every digit in a numeral
has two values:
a. Its face value is equal to it and never changes.
b. Its place value depend on its position in numeral.
1. Natural Number
: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, …… ∞}$(0$excluding)
2. Whole Numbers
: W= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …… ∞} (0 including) .
All natural numbers are whole numbers. 0 is only a whole number and not a
natural number. Aaryabhatt, who was an astronomer and lived in India around
500 AD, invented the symbol 0 for nothing.
3. Even Number
: E= {2, 4, 6, 8, 10 ………………..}
4. Odd Numbers
: O= {1, 3, 5, 7, 9, 11, ……}
5. Positive integers
: I+ = {+1, +2, +3, …….}
6. Negative Integers
: I- = {-1, -2, -3, -4,…………..$}
7. Integers
: I= {…24,23,22,21,0,1,2,3,4…..}
8. Zero
:
a. Zero is an integer and it is neither positive nor negative.
b. Any number multiplied by 0 is equal to 0.
c. Face value and place value of 0 is always 0
9. Imaginary Numbers: The square root of negative numbers are called
imaginary numbers. Example :
,
,
etc.
10. Real Numbers : When we put the irrational numbers together with the rational
numbers, we finally have the complete set of real numbers. Real numbers are
• Rationals + Irrationals
• All points on the number line
• Or all possible distances on the number line
Real numbers include
•
•
•
Whole Numbers (like 1,2,3,4,5,6 etc)
Rational Numbers (like 3/4, 0.1, 0.635..., 1.5, etc )
Irrational Numbers (like π, √3, etc )
What is not a Real number ?
•
•
√-4 (the square root of minus 4) is not a Real Number, it is an
Imaginary Number
Infinity is not a Real Number
The Real Number Line
The Real Number Line is like an actual geometric line.
A point is chosen on the line to be the "origin", points to the right will be positive,
and points to the left will be negative.
A distance is chosen to be "1", and the whole numbers can then be marked off:
{1,2,3,...), and also in the negative direction: {-1,-2,-3, ...}
Any point on the line is a Real Number:
•
•
The numbers could be rational (like 20/9)
or irrational (like$!)
Properties of Real Number :
a. Closure Property : The sum and multiplication of the real numbers
are always a real number. Example : x + y = z and a x b = c, here x,
y, z, a, b and c are integers.
b. Opposite
: Two real numbers that are the same
distance from the origin of the real number line are opposites of each
other.
c. Reciprocals
: Two numbers whose product is 1 are
reciprocals of each other.
d. Commutative Property
: When adding or multiplying two numbers,
the order of the numbers does not matter. Example : a + b = b + a or
a x b = b x a, where a and b are rational numbers.
e. Associative Property
: When three numbers are added, it makes
no difference which two numbers are added first. And at the same
time when three numbers are multiplied, it makes no difference which
two numbers are multiplied first. Example : (a+b)+c=a+(b+c) or
(axb) x c= ax (bxc), where a, b and c are rational numbers.
f. Distributive Property
: Multiplication distributes over addition ie.
ax(b+c)=(axb) + (axc), where a, b and c are integers.
g. Additive Identity Property : The additive identity property states that if
0 is added to a number, the result is that number. Example :
5+0=0+5=5
h. Multiplicative Identity Property: The multiplicative identity property
states that if a number is multiplied by 1, the result is that number.
Example : 5 x 0= 0 x 5=0
i. Additive Inverse
: The additive inverse property states that
opposites add to zero ie. if the sum of two rational numbers is zero the
numbers are known as additive inverse of the other. Example :
+ ( )=0. Additive inverse of every rational number x is –x.
!
!
j. Multiplication Inverse
: The multiplicative inverse property states
that reciprocals multiply to 1 ie. if the multiplication of two numbers is
1, then each number is known as multiplication inverse of the other.
Example : 3 x
= 1 or
x
= 1. Multiplication inverse of every
rational number x will be
11. Composite Numbers : A composite number is a positive integer which has a
positive divisor other than one or itself. In other way a Composite number has
more than two divisors. i.e. 10 is a composite number as it has 1,2,5 and 10
as divisors. Other examples : 4,6,9,16,21,… etc.
When a number can be divided up evenly it is a Composite Number
12. Prime Numbers
:
When a number can not be divided up evenly it is a Prime Number.
A natural number other than 1, which has no other factor except 1 and itself,
is called a Prime Number. Examples : 2,3,5,7,11,13,17,…… are prime numbers.
When only two factors of a number are 1 and the number, then the number is
called Prime Number and it must be greater than 1.
A prime number is greater than 1 and a Positive integer.
Any natural number greater than 1 is either composite or prime number.
List of prime number :
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97 101 103 107 109 113 127 131 137 139 149 151 157 163
167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269
271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383
389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499
503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619
631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751
757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881
883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997
1 is neither Prime nor Composite number.
The smallest prime number is 2.
2 is the only even Prime number, all other Prime Numbers are odd.
13.Co-Prime Numbers : Two numbers are said to be Coprimes when they have
only 1 as a common factor.
Two natural numbers are called co-prime numbers if they have no common
factor other than 1 or two natural numbers are co-prime if their H.C.F is 1.
Example 1
: Find out whether number 7 and 12 are Coprime ?
Answer
:
Factors of 7 are 1 and 7
Factors of 12 are 1, 2,3,4 and 12
On Comparing the factors of number 7 and 12, we find that both have only 1
as a common factor. Hence, the numbers 7 and 12 are Coprimes.
14.Twin Prime Number : The pairs of Prime Numbers whose difference is 2 are
called Twin Prime Numbers. Example : (5,7), (11,13),(17,19) etc.
15.Prime Triplet Number
: The three natural numbers whose H.C.F. is 1, are
called Prime triplet Numbers. Example; (8,9,25).
In mathematics, a prime triplet is a set of three prime numbers of the form
(p, p + 2, p + 6) or (p, p + 4, p + 6). With the exceptions of (2, 3, 5) and (3,
5, 7), this is the closest possible grouping of three prime numbers, since every
third odd number greater than 3 is divisible by 3, and hence not prime.
The first prime triplets are
(5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43),
(41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109),
(107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227,
229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349,
353), (457, 461, 463), (461, 463, 467), (613, 617, 619), (641, 643, 647),
(821, 823, 827), (823, 827, 829), (853, 857, 859), (857, 859, 863), (877,
881, 883), (881, 883, 887)
A prime triplet contains a pair of twin primes (p and p + 2, or p + 4 and p + 6
), a pair of cousin primes (p and p + 4, or p + 2 and p + 6), and a pair of sexy
primes (p and p + 6).
16. Perfect number
: A given number for which sum of all its factors is equal
to it's twice, the number is known as A Perfect Number. Example :
Examples 1 = Is 6 a Perfect number ?
Answer = Find the factors of 6
Factors of 6 = 1, 2, 3, 6.
Add all the above factors.
Sum = 1 + 2 + 3 + 6 = 12
Twice of given number 6 = 12
Since the sum of all the factors of 6 is equal to its twice
So, 6 is a perfect number.
Examples 2 = Is 9 a Perfect number ?
Answer = Find the factors of 9.
Factors of 9 = 1, 3, 9.
Add all the above factors.
Sum = 1 + 3 + 9 = 13
Twice of given number 9 = 18
Since the sum of the factors of 9 is not equal to its twice
Hence, 9 is not a perfect number.
Examples 3 = Is 28 a Perfect number ?
Answer = Find the factors of 28
Factors of 28 = 1, 2, 4, 7, 14, 28.
Add all the above factors.
Sum = 1 + 2 + 4 + 7 + 14 + 28 = 56
Twice of given number 28 = 56
Since, the sum of all the factors of 28 is equal to its twice
So, 28 is a perfect number.
17.Literal Numbers
: An English Alphabet that is used to represent a variable
is called a Literal Number.
18.Fraction
: A fraction is a quantity that can not be expressed in
whole number. A fraction is an ordered number of whole numbers, the first
one is written on top of other, eg. ¼, 3/5,2/7 etc.
The top number is called the Numerator, it is the number of parts one has.
The bottom number is called the Denominator, it is the number of parts the
whole is divided into.
Numbers which can be expressed as
where (a) x
1 (d) there is no common factor. Example:
Equivalent Fraction
, ,
y, (b) x ≠ 0, y ≠0 (c) y≠
etc.
: A fraction can have many different appearances
these are called Equivalent Fraction. Example:
and
and
and
.
We cannot tell whether two fractions are the same until we reduce them to
their lowest terms.
Improper Fraction
: An improper fraction is a fraction with the numerator
larger than or equal to the denominator. Example :
, ,
etc.
Any natural number can be converted into Improper Fraction.
Mixed Number
: A mixed number is natural number and a fraction
together. An improper fraction can be converted to a mixed number and vice
versa.
19. Inverse Numbers
: Numbers who have negative power are called Inverse
Numbers. Example : 3-2 , 4-5 etc.
20. Even Number
: Even numbers can be divided evenly into groups of two(
or any natural number which is divisible by 2 is called even number. Example
:2, 4, 8, 10,22 etc.
21. Odd Number
: Odd numbers can not be divided evenly into groups of
two or any natural number which is not divisible by 2 is called an odd number.
Example : 1, 3, 5, 7, 111 etc.
22.Rational numbers
: Numbers which can be represented as where x and y
are integers and y
0. Eg. 1, 2, -1, -2,
, - !$etc.
Special characteristics of rational numbers:
a. All natural numbers are rational.
b. Rational number can be expressed either as a terminating decimal or a
repeating decimal.
c. Every terminating decimal is a rational number.
d. Every repeating decimal is a rational number.
23. Irrational Number : Numbers which can not be represented as ratio of two
natural numbers, are called irrational number.!Example : , 5
etc.
An irrational number is a non repeating and non terminating decimal.
! and ! are irrational numbers.
History of Irrational Numbers
Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers
when trying to represent the square root of 2 as a fraction (using geometry, it is
thought). Instead he proved you couldn't write the square root of 2 as a fraction
and
so
it
was
irrational.
However Pythagoras could not accept the existence of irrational numbers, because
he believed that all numbers had perfect values. But he could not disprove
Hippasus' "irrational numbers" and so Hippasus was thrown overboard and
drowned!
24.Decimal form of a rational number
: When we express a rational number
in decimal, then either the decimal will be exact, as = 0.5 (called terminating
decimal) or it will not, as
= 0.333…. (called recurring decimal).